Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(10,15)(12,17)(16,19)(18,24);(1,2,3,4)(9,10,13,15)(11,18,20,24)(12,22,17,14)(16,23,19,21);(6,8)(9,11,13,20)(10,16,15,19)(12,18,17,24)(14,23,22,21);(9,12,13,17)(10,14,15,22)(11,19,20,16)(18,23,24,21);(10,17)(12,15)(14,22)(16,24)(18,19)(21,23)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283560, #72283612, #72284761, … |
| maximal normal subgroup of | #72283560, #72283612, #72284761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481116);